54.3.5 problem 5

Internal problem ID [8561]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 5
Date solved : Wednesday, March 05, 2025 at 06:09:38 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-4 x^{2}+1\right ) y^{\prime \prime }+8 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 39

Order:=8; 
ode:=(-4*x^2+1)*diff(diff(y(x),x),x)+8*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (-4 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{3}-\frac {16}{15} x^{5}-\frac {64}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 40

ode=(1-4*x^2)*D[y[x],{x,2}]+8*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (1-4 x^2\right )+c_2 \left (-\frac {64 x^7}{35}-\frac {16 x^5}{15}-\frac {4 x^3}{3}+x\right ) \]

✓ Sympy. Time used: 0.840 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 4*x**2)*Derivative(y(x), (x, 2)) + 8*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (1 - 4 x^{2}\right ) + C_{1} x \left (- \frac {16 x^{4}}{15} - \frac {4 x^{2}}{3} + 1\right ) + O\left (x^{8}\right ) \]